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A new method for solving the 1D Poisson equation is presented using the finite difference method. This method is based on the exact formulation of the inverse of the tridiagonal matrix associated with the Laplacian. This is the first time that the inverse of this remarkable matrix is determined directly and exactly. Thus, solving 1D Poisson equation becomes very accurate and extremely fast. This method is a very important tool for physics and engineering where the Poisson equation appears very often in the description of certain phenomena.

The finite difference method is a very useful tool for discretizing and solving numerically a differential equation. It is effectively a classical method of approximation based on Taylor series expansions that has help during the last years theoretical results to gain in accuracy, stability and convergence.

In fact, this method is very useful for solving for example Poisson equation. This elliptic equation appears very often in mathematics, physics, chemistry, biology and engineering. In one dimension, the resolution leads to a tridiagonal matrix in the case of centered difference approximation. This matrix, which is diagonally dominant, can be inverted with methods such as Gauss elimination, Thomas Algorithm Method [

We proposed here, a new and direct method of inversion of this tridiagonal matrix independently of the right- hand side. For Dirichlet-Dirichlet boundary problems, this innovative method is faster than the Thomas Algorithm. It gives better accuracy and is far more economical in terms of memory occupation.

First, the finite difference method is presented for the 1D Poisson equation. Secondly, the properties of the matrix associated with the Laplacian and its inverse are discussed. Then, the inverse matrix is determined and its properties are analyzed. Thus, verification is done considering an interesting potential problem, and the sensibility of the method is quantified.

We consider a function

fined by

We have chosen the centered difference approximation

One gets a linear system of

Thus, solving the 1D Poisson equation means to invert the negative definite, and regular

and

where

From (4), we derive successively the following interesting relations:

with (5), one sees that the matrix

From (5) and (6), we get

Now, the matrix

The solution of the 1D Poisson equation is obtained with a simple, extremely fast matrix multiplication:

A first analysis of the matrix

A deeper analysis of the matrix

The matrix

All the information about it, can be found in the upper triangle (in gray color, see

Further, we can even find very interesting relations in this matrix which can help refining the final solution.

That is what we effectively did, and one can see a direct solution for

Also a direct solution for

Generally, a very important recurrence relation can be obtained, which gives all solutions:

which is equivalent to:

This very innovative Equation (12) gives directly and accurately all the solution that we are looking for. It proves that our method is direct, faster than the one of Thomas’s in this context and gives as well better accuracy. Furthermore, it is far more economical in terms of memory occupation. This is due to the fact that the matrix

In conclusion to this, we can say that the matrix

N.B.: One can prove using mathematical induction that

We consider a scalar potential

With the finite difference method, we take

We define the variable

Generally, we have

We can also define the average value of the relative error for a given

We obtain the following results, presented in

The table shows that the solution is very accurate. Notwithstanding that we have been interested in determining the sensibility of the proposed method. Effectively, we have plotted

We obtain a hyperbola, which can be predicted as proportional to

This curve is fitted with a function which can be defined as

where

We realize that the average relative error

lowing manner

For our given function

This proves that the method is very accurate, naturally stable, robust, quick and precise.

This paper has provided a new improved method for solving the 1D Poisson equation with the finite difference

method. Accurate results have been obtained with a sensibility found to be as the function of

A future work can consider Neumann or mixed boundary conditions.

I would like to thank my colleagues Dr. Cheikh Mbow and Dr. Kharouna Talla for benefit discussions and remarks that contribute to improving the quality of this paper.